Theorem prnz 4748

Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Hypothesis

Ref	Expression
prnz.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
prnz	⊢ {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz

Step	Hyp	Ref	Expression
1		prnz.1	. . 3 ⊢ 𝐴 ∈ V
2	1	prid1 4733	. 2 ⊢ 𝐴 ∈ {𝐴, 𝐵}
3	2	ne0ii 4300	1 ⊢ {𝐴, 𝐵} ≠ ∅

This proof depends on syntax axioms:   ∈ wcel 2146   ≠ wne 2961  Vcvv 3458  ∅c0 4289  {cpr 4596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-v 3460  df-dif 3911  df-un 3913  df-nul 4290  df-sn 4595  df-pr 4597

This theorem is used by:  opnz  5460  propssopi  5496  fiint  9296  wilthlem2  27270  upgrbi  29480  wlkvtxiedg  30011  shincli  31751  chincli  31849  constrextdg2lem  34169  spr0nelg  48265  sprvalpwn0  48272